Step 1 amounts to solving the polynomial system given by \(A \, x = b\)  and \(x \in T_{A,c,\epsilon }\) .
For linear programming, one wants the unique positive solution \(x^*(\epsilon )\) .
But, for other applications, e.g. scattering amplitudes in particle physics [1]},
all complex solutions are needed. A standard method for finding them all
is homotopy continuation [2]}.
We expect the number of solutions to be
\(\deg T_{A,c,\epsilon } = {\rm vol}({\rm conv}(A \cup 0))\) , and this
is the number of paths to be tracked. This number is also the algebraic degree
of \(x^*(\epsilon )\) , over the ground field \(\mathbb {Q}(z)\)  in (REF ).
